The equation describes a circle with the center at the origin (0,0) and a radius of 1.

Above, x and y are raised to a small even power. Now we will look at the graph when x and y are raised to a higher even power. Notice that the circle begins to transform into a square. Also, as the powers increase, the figure becomes more square like and the edges are more pointed.

What happens when the x and y are raised to an odd power such as ? The graph appears to be a line with one corner of the square.

Look at the graph when x and y are raised to a higher odd power. The corner of the square appears to become more pointed and the line approaches y=-x as x and y increase.

Now look at the graph when you add a coefficient for x into the equation. The equation appears to be an ellipse.

Look at the graph when the coefficient of x is increased and decreased. As the coefficient decreases, the figure resembles an ellipse. As the coefficient increases, the figure becomes more rectangular shaped. Notice that the height and the center of the shape remain unaffected in all of the equations...only the width changes.

Now look at the graph when you add a coefficient for y into the equation. Similar to altering the coefficient of x from above, as the coefficient of y decreases, the figure resembles an ellipse. As the coefficient increases, the figure becomes more rectangular shaped. However, the width and center of the shape now remain unaffected...only the height changes.

Conclusion:

Given the equation , we saw that the graph was a circle with center (0,0) and a radius of 1. By raising the x and y to a larger even power we noticed that the figure becomes a square with more pointed edges. By raising the x and y to an odd power, we noticed that the figure appears to be a line and one corner of a square. As the odd powers of x and y increase the corner becomes more pointed and the line approaches y=-x. We then looked at the graph when you add coefficients for x and y. If we increase or decrease the coefficient of x, the width changes...the height and center of the shaped remain unaffected. If we increase or decrease the coefficient of y, the height changes...the width and center of the shape remains the same.